For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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Proof of Jordan Theorem on Lie Algebra

$ L$ is semisimple Lie algebra in ${\rm gl}\ V$ where $V$ is a complex vector space. Then we have two Jordan forms $$ x=d+n,\ {\rm ad}_x = {\rm ad}_d + {\rm ad}_n\ (x\in L) $$ (cf. Theorem 9.15 in ...
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135 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
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84 views

Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...
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125 views

Vector fields on smooth manifolds and Lie algebras

I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to ...
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88 views

Open questions in theory of Lie groups and Lie algebras

I am just about to finish an introductory book 'Lie groups, Lie algberas & Representations' by Brian Hall and am curious to know what are the current directions of research in this area. I learn ...
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65 views

How can I show that $\mathfrak{sl}_n(\mathbb{C})$ is a simple Lie algebra?

The question is in the title: how can I show $\mathfrak{sl}_n(\mathbb{C})$ is simple? In every book I scoured, they say $\mathfrak{sl}_n(\mathbb{C})$ is simple but they do not provide a proof! Is ...
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69 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
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66 views

Dynkin diagram construction

My question is how to construct the Dynkin diagrams of a semi-simple Lie group $G$, which is the product of simple Lie groups. Is it the combinaison of Dynkin diagrams of these simple Lie groups? For ...
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32 views

Example showing that the product of ideals must be the span of the commutators

I'm trying to find an example showing why, in a Lie algebra, we can't just define the product of two ideals $I$ and $J$ to be the elements of the form $[x,y]$ where $x \in I, \; y \in J$. I imagine ...
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98 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
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46 views

Relationship between invariants of a simple algebraic group

Let $G$ be a simple algebraic group over an algebraically closed field $k$. I believe all of the following invariants are well-defined. Besides the coxeter number, I haven't read about the others, ...
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36 views

explicit realization of irreducible representations of simple lie algebras

I know explicit realization of irreducible representations of simple lie algebra $sl_n$ when the highest weight of that representation is a fundamental weight.Is there any explicit realization of any ...
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112 views

$\mathfrak{sl}(3,F)$ is simple

Prove that $\mathfrak{sl}(3,F)$ is simple, unless $\operatorname{char}F=3$. [Use the standard basis $h_1,h_2,e_{ij}(i\neq j)$. If $I\ne 0$ is an ideal, then $I$ is the direct sum of eigenspaces for ...
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75 views

Does the Lie Bracket automatically exist?

Let $g$ be a Matrix Lie Group. The Lie Algebra of $g := Lie(g)$ is defined as $ Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1, \gamma(0) = \mathbb{I} \} $ ...
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126 views

Root-subspaces are $\mathfrak{g}$-invariant

I need some help with technicalities concernig root-subspaces of nilpotent Lie algebras of operators. Let $\mathfrak{g} \subseteq \mathfrak{gl}(V)$ be nilpotent Lie algebra, $\alpha \ \colon \ ...
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33 views

Classification of 6-D Nilmanifolds

I am reading the G.Cavalcanti and M.Gualtieri's Generalized Complex Structures on Nilmanifolds. In the introduction it is said that there are 34 nilpotent lie algebra isomorphism classes. There are ...
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70 views

Cartan's Criterion. $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$

Cartan's Criterion. Given $V$ a finite dimensional complex vector space and $L$ a Lie subalgebra of $gl(V)$ then, $L$ solvable $\implies$ $Tr(xy) = 0$ $\forall x \in L$, $\forall x \in L^{(1)}$. ...
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46 views

Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
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67 views

Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
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166 views

Semi-simple Lie algebra $L$ coincides with its derived algebra $L'$

If $L$ is a semi-simple Lie algebra, then $L=L'$. Since $L$ is semi-simple we can write it as a direct sum of simple ideals $L_i$, i.e. $L=\oplus_{i=1}^r L_i$. Then $L'=\oplus_{i=1}^r L_i'$ and ...
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37 views

To what extent are formulas obtained in one Lie group valid in another Lie group with an isomorphic Lie algebra?

In quantum optics, I am trying to explore the group generated by squeezing and rotation operators. These are closely related to area-preserving linear transforms, which they induce on the phase space, ...
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89 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
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162 views

Determinant of a cartan matrix

I was taking an introductory course in Lie algebras and I just learned about how we associate a Cartan matrix to a semisimple Lie algebra. So, for the A-series, the determinant of this matrix goes to ...
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24 views

Finding the Lie map

Suppose I have a group homomorphism $\rho:SL(2,\mathbb{C})\to SO_0(3,1)$ given by $\rho(a)X=aXa^*$ and I want to see how the corresponding Lie map $L\rho$ looks like. By definition $$ ...
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32 views

Only 3 finite-dimensional Lie algebra on $\mathbf R$?

Please, how does one show that up to diffeomorphism there are exactly three finite dimensional Lie algebras of vector fields on the real line $\mathbf R$.
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98 views

If $\mathfrak{g}$ admits a decomposition then it is semisimple

I want to show: If $\mathfrak{g}$ is a Lie algebra that has an abelian subalgebra $\mathfrak{h}$ such that $\mathfrak{g}$ has a Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus(\bigoplus_{\alpha ...
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24 views

Integer domain enveloping algebra

I must prove that if $L$ is a Lie algebra and denoting $U(L)$ the enveloping algebra, then $U(L)$ hasn't zero divisions (e.g. if $ab=0 \,\,\, a,b \in U(L)$ then $a=0$ or $b=0$). Some ideas?
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79 views

$x$ regular $\Leftrightarrow$ $x$ is in exactly one CSA

Here's a statement and a proof given in a Lie Algebra course (in the tutorial): Let $L$ be a semisimple Lie algebra over a field $F$ with $\text{char} F=0$. Let $x\in L$ be a semisimple element. ...
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62 views

Finite-dimensional Lie algebra as a scheme

Kindly asking for any hints about the following questions: Suppose $k$ is an algebraically closed field of characteristic zero and $g$ is a finite-dimensional Lie algebra over $k$. Then $g$ is ...
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76 views

Dimension of Abelian Lie Algebras

I've tried to answer this question, but I need some help. What is the possible dimension of irreducible representations of Abelian Lie Algebras? I think it is always one, but I am not sure. Thank ...
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89 views

Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ ...
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38 views

Smallest dimensional irreps of semi-simple Lie algebras

I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
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240 views

Standard parabolic Lie subalgebras and conjugacy

Let $\mathfrak g$ be a given semisimple Lie algebra with corresponding adjoint Lie group $G$. A parabolic subalgebra is any subalgebra containing a Borel subalgebra. We can pick a Borel ...
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ADE type root lattice

Let $\Phi$ be a root system of ADE type, $L$ is the corresponding root lattice, show that $\Phi=\{\alpha\in L:(\alpha,\alpha)=2\}$, where $(,)$ is the normalized non-degenerate symmetric bilinear form ...
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252 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
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68 views

Basis of the Engel algebra

If I have a connected, simply connected nilpotent lie group given by the commutators between the elements of a basis of its Lie algebra how can I recover the left invariant vector fields? For ...
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130 views

Finding the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$

How can I find the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$? I'm explicitly working with basis vectors in trying to compute $\operatorname{tr}(\operatorname{ad}(a)\operatorname{ad}(b))$ but ...
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66 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
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111 views

Question about a Corollary of Engel's Theorem

Engel's Theorem states that: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V \neq 0$, then there exists nonzero $v \in V$ for ...
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78 views

Principal Bundles and Lie algebras

Suppose I have a principal $S^{1}$ bundle over a nice compact symmetric space. The symmetric space arises as a homogeneous space, call it $X=G/H$. On the Lie algebra level we have the decomposition ...
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43 views

Solve commutator relation $[Q,d]=-[P,d]$ for $Q$ on chain complexes with scalar product

Suppose we are given chain sequences $\dots \rightarrow C_k \rightarrow C_{k+1} \rightarrow \dots$ and $\dots \rightarrow D_k \rightarrow D_{k+1} \rightarrow \dots$ of finite-dimensional vector ...
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76 views

orbit of a Dynkin diagram automorphism

Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$. What is a set of representatives of the orbits of $\Delta$ under $f$ ? Thanks,
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Is it possible to chose the value of $(\alpha, \alpha)$ for the root system $A_{1}$?

I am looking at the Lie Algebra $L=\mathfrak{sl}(2, \mathbb{F})$ with the root system $A_{1}=\{-\alpha, \alpha\}$. $char\mathbb{F}=0$. $( \ , \ )$ is a scalar product. Let ...
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$Z(\mathcal{L}^{(n)}) \subset Z(\mathcal{L})$ for solvable Lie algebras?

$X$ Banach space. $\mathcal{L} \in B(X) $ is solvable Lie Algebra. Then for some n, $\mathcal{L} \supset \mathcal{L}^{(1)}=[\mathcal{L},\mathcal{L}] \supset ...
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19 views

How do I know the length of a Weyl group element from its Weyl orbit result

How do I know the length of a Weyl group element from its Weyl orbit result? For example, I know that $[2,2]$ under $s_1s_2s_1$ transforms into $[-2,-2]$,but given the result, how can I tell ...
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40 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
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22 views

Lie algebra operations from lie group

According to wikipedia, if $G$ is a closed subgroup of $GL(n, \mathbb{R})$ then the Lie algebra of $G$ can be thought of informally as the matrices $m$ of $M(n, \mathbb{R})$ such that $1 + εm$ is in ...
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The algebra of $W$-invariant polynomial funktions sl2

Let $L=\mathfrak{sl}(2,\mathbb{F})$, $H$ a borel subalgebra, $\Delta=\{\alpha\}$ a base of the corresponding root system and $W$ the Weyl group. Let $\lambda=\frac{1}{2}\alpha$ be the fundamental ...
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31 views

Universal enveloping algebra of sl2

I am currently trying to proof, that $x-1$ is not invertible in the universal enveloping algebra $\mathfrak{U}(\mathfrak{sl}(2,\mathbb{F}))$ of $\mathfrak{sl}(2, \mathbb{F})$, but I still struggle ...